Apologies if someone has already mentioned this, but I found this section to be missing from the audio: "As we counsel young men concerning military service, we must clarify for them our nation’s role in Vietnam and challenge them with the alternative of conscientious objection. [sustained applause] I am pleased to say that this is a path now chosen by more than seventy students at my own alma mater, Morehouse College, and I recommend it to all who find the American course in Vietnam a dishonorable and unjust one. [applause] Moreover, I would encourage all ministers of draft age to give up their ministerial exemptions and seek status as conscientious objectors."

As others have noted this isn't complete, go here for complete or to others noted below:
http://mlk-kpp01.stanford.edu/index.php/encyclopedia/documentsentry/doc_beyond_vietnam/
I disagree that it is worth cutting it short, the full thing is not that much longer and is, simply, complete.

http://www.informationclearinghouse.info/article2564.htm
Audio includes King's own introduction. Site also has the text.

This recording contains MOST of MLK's speech but not all of it. Several minutes have been cut. Among the deleted passages are the introductory acknowledgements and 4 of the 5 specific steps MLK proposes for ending the war in Vietnam.

To me, these cuts do not look malicious. It looks like, years later, some material that looked less essential to a later listener was edited out--perhaps to make the speech fit into a radio timeslot.
Here is what looks like a good, complete written transcription of the speech: http://www.americanrhetoric.com/speeches/mlkatimetobreaksilence.htm . I don't know where a complete audio recording may be found (possibly the King Archives).

We have faltered upon the words of the greater, and of all that Martin Luther King Jr. did, that man of greatness arose to bring us forth unto the light of a brighter day... a day perennial... in which shall be shone all that is and shall be. My review isn't necessary. Not praise either, for what shall be said shall be said through our being. We must be carried forth by this endless way… Dr. King you are a beautiful man whose truth I shall know from within, and understand from beyond. You lead me, and so you shall lead all. Vietnam seems to be of a time in our history where quick was the ripening of American fruit… it seemed so sweet till rotten it became, and we knew not the failure we committed ourselves to… extend atrocities by a vulgar ugliness rampantly crowding the air with an American noise- the shouts of maddened patriotism and of imperialist blindness. When shall our eyes awaken I cry, only to hear you answer Dr. King

Minister King demonstrates that whites really are threatened by people with a genuince conviction to eradicate social injustice.
King was killed because he became serious. Let us not forget that
as for Tavis Smiley, didn't he interview Tim Wise???

Thanks for posting this, such an important speech. As Tavis Smiley said, "King was gunned down on that balcony in Memphis a year to the day later...after that speech, he was persona non grata in this country. Over 55 percent of black Americans had turned against their leader, Dr. King. Three-quarters of the American people—three-quarters of the American people—had turned against Dr. King. He was no longer on the list of the most admired Americans. So King dies, again, persona non grata on that balcony in Memphis."
Many seem to forget that. Where are the leaders who have this kind of courage today?